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Theorem ndmovcom 6174
Description: Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.)
Hypothesis
Ref Expression
ndmov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )

Proof of Theorem ndmovcom
StepHypRef Expression
1 ndmov.1 . . 3  |-  dom  F  =  ( S  X.  S )
21ndmov 6171 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  (/) )
3 ancom 438 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
41ndmov 6171 . . 3  |-  ( -.  ( B  e.  S  /\  A  e.  S
)  ->  ( B F A )  =  (/) )
53, 4sylnbi 298 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( B F A )  =  (/) )
62, 5eqtr4d 2423 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  ->  ( A F B )  =  ( B F A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   (/)c0 3572    X. cxp 4817   dom cdm 4819  (class class class)co 6021
This theorem is referenced by:  addcompi  8705  mulcompi  8707  addcompq  8761  addcomnq  8762  mulcompq  8763  mulcomnq  8764  addcompr  8832  mulcompr  8834  addcomsr  8896  mulcomsr  8898  addcomgi  27330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-dm 4829  df-iota 5359  df-fv 5403  df-ov 6024
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